While not limited to the field of determining the effect of nuclear irradiation on the mechanical properties of materials, the impetus for the invention originated with the needs and necessities of this activity. The invention is fully applicable to the determination of mechanical behavior of materials not subjected to radiation, and the validity of the invention was demonstrated for materials not subjected to radiation.
Determination of the mechanical behavioral physical properties of materials is very necessary so that the materials may be selected for use and evaluated when in use. From these determinations, decisions are made as to which materials to use, the conditions under which they can be used, and whether materials in use can be continued with safety.
In the past, the most common procedure has been to determine the mechanical behavior of a material by testing large samples that are created more or less simultaneously or "side by side" with the product that is intended to be used. In the determination of the mechanical behavior of solid materials, and particularly metals, the practice is to make tensile, S-N fatigue, creep, stress relaxation, ductile/brittle transition, compact tension, fatigue crack initiation/propagation, fracture modes, fracture stress/strain, multilayered specimens, residual plastic stress/strain, ion irradiated, etc. specimens, and these are then subjected to forces while measurements are taken of the force, time, displacement, impact energy decrement, velocity, etc. of the specimen. Information on stress and strain, which can be thought of as normalized load and deflection respectively, are then obtained by simple mathematical operations. For example, in a uniaxial tensile test, the stress is determined by dividing the measured load by the specimen cross sectional area.
While this may be satisfactory in most instances, there are other circumstances, such as the post-irradiation testing of materials used in nuclear reactors, where samples are unavailable in sufficient size and quantity to carry out these destructive tests during the life of the materials in use. In general, neutron irradiation space for materials investigation is limited and costly. It is therefore desirable to use specimens of minimum volume. Since neutron irradiation costs scale with specimen volume, miniaturized mechanical behavior testing can provide significant savings in irradiation testing costs for nuclear materials investigations. In addition, it is possible to provide mechanical behavior information which is not ordinarily obtainable due to space limitations in irradiation experiments, and thus expedite alloy development investigations. Of course, miniature specimen testing is applicable to materials investigations for other nuclear technologies as well as non-nuclear technologies requiring mechanical behavior characterization from a small volume of material.
While the phenomenon of nuclear radiation on materials is complex, it is well known that materials change in various properties, often drastically, when irradiated. The materials used in nuclear reactors must be pretested under simulated inservice conditions, developed to an optinium design state, and often further tested while in service.
The tensile behavior of a material as the term is used herein is determined from the stress/strain curve measured on the material when subjected to various processes of loadings. The stress/strain curve for a material is most often determined by gripping a large specimen at opposite ends and subjecting the specimen to tension while measuring the load and displacement as a function of time. Since the forces can be high, there is a practical minimum limit to the size of the specimen, as there must be material available for testing, gripping, and there must be room for the apparatus to perform the gripping function. These considerations also apply to other conventional mechanical behavior tests such as fatigue, creep, stress relaxation, ductile/brittle transition, compact tension, etc.
The present invention was conceived as a solution to the problem of determining mechanical behavior from specimens which are smaller in size than the conventional test specimens. There are three principal conceptual innovative aspects to the miniaturized bend test (MBT) of this invention. The first is the use of specimens that are significantly smaller than those currently in use or that are significantly smaller than the in-service components from which they are cut. The second is the use of the appropriate loading configuration to either accommodate the size scale involved or better represent the actual in-service loading. In practice, bending is used to extract mechanical behavior information from a very small sample as opposed to the more standard approach of using uniaxial tension/compression loading requiring gripping extensions. The third is the use of the finite element method to extract useful engineering information from the experimental data.
Others have made suggestions in this field and their publications are listed further in this specification. Their publications and the Thesis cited in the next paragraph are incorporated herein by reference as fully as if they were presented in complete text.
This invention is described in further complete detail in the Thesis entitled, "The Development of a Miniaturized Disk Bend Test for the Determination of Post-Irradiation Mechanical Behavior", by Michael Peter Manahan (an applicant herein),--submitted to the Department of Nuclear Engineering in Partial Fulfillment of the Requirements for the Degree of Doctor of Science at the Massachusetts Institute of Technology--May 1982 (1).
The determination of a stress/strain curve, using analytical expressions from a pure bending test with large specimens, was first reported by Herbert (2) for cast iron bars. More recently, Crocker (3) has obtained stress/strain information for large specimens with large deflections and small strains using a three point rotary bend test. He used the same analytical expressions as Herbert and implemented a progressive reconstruction technique to transfer the moment-angle measurements into a stress/strain relationship. Stelson et al. (4) have used an adaptive controller to measure force and displacement during brakeforming of large components to estimate workpiece parameters with a microcomputer, which are then used in an analytical elastic-plastic material model to predict correct final punch position. Although the earlier developments have been useful, particularly in the metals forming industry, they are not readily adaptable to miniaturized mechanical behavior testing because of large specimen size and awkward loading configuration. A coarse ductility screening test for miniature disks, with very small ductility, has also been developed (5) using elastic analytical equations. The earlier developments have not recognized or suggested the advantages of bend testing of miniature specimens of a size at or close to the limit of continuum behavior in the material in all directions. The advantage of the finite element method for data inversion in the MBT is that it permits the extraction of both plastic resistance and creep resistance from the raw data in addition to the information on ductility from irradiated samples exhibiting moderate to large levels of strain to fracture and with a minimum of material.
Finite element analysis is performed to convert the experimental central load/deflection curves into stress/strain and other useful engineering information. In order to accurately analyze the MBT using the finite element method, a new finite element frictional contact boundary condition model has been developed (1). The strain field present in MBT is, in general, highly non-uniform throughout the sample unlike the more conventional uniaxial tensile strain fields which are constant (in the gage section) for a given static load up to the point of plastic instability. Therefore, accurate three-dimensional boundary condition modeling is essential in simulating the actual strain gradients in the specimen during the experiment. The model accounts for this highly non-linear boundary value problem with shifting frictional contacts.
The MBT problem contains all three types of non-linearity that can be encountered in stress analysis; namely, material, geometric, and boundary. The first two classifications of non-linearity have been adequately addressed in several general purpose finite element codes (6, 7). The latter classification of non-linearity has not been adequately addressed to date, and therefore a new finite element friction-gap boundary condition model has been developed. Although the model has been applied to the MBT problem in particular, the method developed is of general applicability to a wide variety of boundary condition problems.
Various methods to deal with friction in the finite element method have been proposed during recent years, primarily for application in the metalforming industry. Nagamatsu et al. (8,9) have introduced a slip-factor which is used to modify surface nodal displacements. Gordon and Weinstein (10), Iwata et al. (11) and Odell (12) have imposed surface nodal forces which are oppositely directed to the nodal displacement direction. A similar technique for opposing surface nodal displacement has been proposed by Shah and Kobayashi (13) and Matsumoto et al. (14) by introducing a surface shear stress which is evaluated from an empirical constant. As formulated by these authors, both the force method and the surface shear stress method are only applicable to problems where the direction of nodal displacement is known prior to the start of the analysis. Sharman (15) and Zienkiewicz et al. (16) have proposed interfacial friction elements for very small deformations which do not require prior knowledge of the direction of nodal displacemented, and utilized them successfully for very small deformations. Hartly et al. (17) have extended this idea to include large deformations. They found that for relatively small deformations the interfacial element layer exhibits unstable deformation and subsequently collapses. They circumvented this difficulty by introducing an element layer stiffness modifying function which depends on the ratio of the yield stress of the surface layer to that of the bulk material. The technique was applied to ring compression and satisfactory results achieved. Although this method appears promising for simple geometries, the validity of this approach has yet to be demonstrated for complex loading geometries. Also, this method, as currently formulated, does not account for shifting contact during the deformation process.
Thus, it is obvious that a new finite element friction model development, which does not depend upon prior knowledge of the deformation kinematics and accounts for shifting contact, is necessary. The model developed herein satisfies these criteria and also requires no kinematic assumptions other than the external boundary condition constraints.
The ABAQUS (6) computer code was chosen for this modeling application because of its superior non-linear capabilities. Some of the more important capabilities necessary to adequately model the MBT non-linear boundary value problem are as follows:
1. two dimensional axisymetric continuum elements PA1 2. multi-linear material hardening PA1 3. large rotations/large displacements PA1 4. finite strains
At the time of the work described herein, ABAQUS had all of these capabilities with the exception of finite-strain theory. (This capability has since been added.) The code implemented small-strain theory with large rotations and large displacements. Rodal has compared finite-strain theory with small-strain theory (18). He compared these two formulations for thin structures (such as beams, rings and plates) and concluded that large differences between the finite-strain theory and small-strain theory results exist for strains greater than approximately 5.0% or larger. These differences were found primarily at regions where large strain gradients occur. However, for high fluence post-irradiation materials investigations at elevated temperatures the small-strain theory may prove adequate in many instances since the ductility of many materials under these conditions is reduced.
Another very important aspect of the ABAQUS code is the fact that it contains a simple two body dual node friction model applicable to cartesian space. The code uses classical Coulomb friction with a stiffness in stick method to aid convergence. This simple model can be used as a basic building block to accurately represent multiple node frictional contact boundary conditions for essentially any geometry by the introduction of the shadow node concept. This theory enables mapping of the region of contact between a support and a deforming structure in contact with it from two dimensional cylindrical space, for example, to two dimensional cartesian space where the code can solve the friction problem. The method is of general applicability.
In essence, two fictitious shadow nodes are introduced into the analysis, somehwere in cartesian space, for every real physical node in the plate that is a potential contact/friction node. One of the shadow nodes models the plate while the other models the deforming structure. Multi-point constraint equations are written to eliminate the plate shadow node degrees of freedom. In this fashion, the friction-gap problem is effectively mapped from two dimensional cylindrical space to two dimensional cartesian space where the code can model two body dual node friction. Since the method operates directly on the plate nodes, it can therefore be termed a direct boundary condition method as opposed to the indirect methods which use interfacial elements. The method is implemented in such a way that the friction forces always oppose the direction of nodal displacement since we map the slip displacements to cartesian space as well. Therefore, when a node changes direction, the nodal surface force automatically changes sign. Also there are no kinematic assumptions on the deformation. Therefore, a solution correct to within the limitations of continuum mechanics is obtained. Phenomena such as separation of the punch and plate near the center are automatically taken into account in this model.
Eight-noded two dimensional axisymetric continuum elements were used for analysis of the MBT experiment. Isotropic hardening with a von Mises yield function was used in all analyses. In the isotropic formulation, the code requires the tangent modulus. Therefore, the uniaxial work hardening curves were multi-linearized in such a way that the energy of the work hardening curves remains approximately constant. Since all elements in the plate are initially rectangular, advantage of superconvergence is taken by using reduced integration.
Limit analysis studies were performed to test out the friction-gap model. The support model was activated and an elastic solution performed for a point loaded plate. The 20 element mesh, which consists of 2 elements through the thickness and 10 elements along a radius, was used. The true plate response lies between two bounds: (1) a roller support which corresponds to a friction coefficient of zero; (2) a fixed node support which corresponds to an infinitely large coefficient of friction. The results using the friction-gap support model with zero and infinitely large friction coefficients were identical to the results obtained using roller support and fixed node support boundary conditions respectively. The friction-gap model limit analysis results using ABAQUS were also compared with the ADINA (7) code results for support boundary conditions with zero and infinitely large friction coefficients for a point loaded plate and were found to be essentially identical.
The friction coefficient for clean stainless steel on clean high density alumina lies between 0.2 and 0.6 (19), and a value of 0.4 was used in all MBT analyses. The mean coefficient of friction has been shown to be approximately temperature insensitive for temperature variations which merely affect the mechanical strengths of the two bodies (20). This is because the ratio of the shear strength to hardness of the weaker material in contact are affected to about the same degree. Since tht MBT testing can be done in inert atmosphere, to first order, the assumption of no temperature dependence of the friction coefficient is valid.
The next step in friction-gap boundary condition model verification was, of course, to activate both the punch and support models and run an elastic-plastic analysis and compare with the MBT data. This has been done and excellent agreement between finite element prediction and experimental data has been observed.
A mesh refinement study was performed to verify that the 20 element mesh is sufficiently refined. A 100 element mesh was run and the solution compared with the 20 element mesh solution. The 100 element mesh consisted of 4 elements through the thickness and 25 along a radius. The results were essentially identical away from the punch. Near the punch, the solution differed somewhat because the boundary conditions were different. The 100 element model is inherently less stiff and also has more potential friction nodes per unit surface length. This results, in general, in more punch surface contact with the plate. There were some slight differences in the central load/deflection response. The 20 element mesh was judged adequate from a mesh refinement standpoint and was used in all subsequent analyses.